In 1832 Hamilton predicted conical refraction, concluding that if a beam propagates along an optic axis of a biaxial crystal, a hollow cone of light will emerge. Shortly afterwards, Lloyd observed the hollow light cone using a natural biaxial crystal and sunlight. Modern studies refer to conical refraction as conical diffraction since its theoretical description requires the inclusion of wave effects, these terms are synonymous.
The optical effects and devices based on conical refraction (CR) phenomenon are of fundamental and practical importance in the field of photonics since most of the known crystal structures are optically biaxial. Yet there are relatively few studies of the phenomenon available. Recent interest is driven by the availability of modern crystal growth, cutting and polishing technologies having advanced to a stage where producing crystals with the correct orientation is now possible. Observation of the CR phenomenon is made using the apparatus as shown in FIG. 1 which comprises a laser 3, a lens 5 and an optically biaxial CR crystal 7 which is cut perpendicular to one of its optic axes. The spatial evolution of an incident Gaussian beam and its transformation under the effect of CR is shown in FIG. 1a. The light ring is observed at the Lloyd plane 19, which is also called the focal image plane. After the Lloyd plane, the beam then progresses to a series of rings first observed by Poggendorff 21, before evolving to an axial spike first noted by Raman 23.
Finally, the beam returns to the original profile in the far field. The Lloyd plane is also a symmetry plane 25 (FIG. 1b). The centre of the ring in the Lloyd plane is laterally shifted by an amount, denoted here by C, which depends on the crystal length, d, and a factor representing the crystal's ability for conical refraction (FIG. 1a). The direction of this lateral shift can be defined as a property of the crystal orientation. A pseudovector, Λ, can also be empirically defined as being perpendicular to both the beam propagation direction and the direction of the lateral shift obeying a right hand rule. Another feature is related to the longitudinal shift of the Lloyd plane. The longitudinal shift, Δ in FIG. 1a, is given by
                    Δ        =                              d            ⁡                          (                              1                -                                  1                  n                                            )                                .                                    (        1        )            
Here n is the refractive index of the crystal in the propagation direction of the photons.